Binomial Distribution & Hypothesis Testing: The Sign Test

Trees Grown

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No Fertilizer

Fertilizer

Decision

The formal steps of hypothesis testing:

Step 1: State your hypotheses.

Step 2: Find the critical value.

Step 3: Calculate the obtained statistic.

Step 4: Make a decision.

Hypotheses: Types and Tails

You must always report TWO hypotheses:H0: The null hypothesis.H1: The alternative hypothesis.

(outcome due to chance alone)

(specific outcome due to IV)

Hypotheses: Types and Tails

Hypotheses can be ONE or TWO-tailed:This coin is weighted for tails.This coin is weighted.

(directional)

(nondirectional)

Example: Two-Tailed. “I think this coin is weighted”

H0: This coin is not weighted.H1: This coin is weighted.

Example: One-Tailed. “I think this coin is weighted for tails”

H0: This coin is not weighted for tails.H1: This coin is weighted for tails.

Example: One-Tailed. “I think this coin is weighted for heads”

H0: This coin is not weighted for heads.H1: This coin is weighted for heads.

Example: One-Tailed. “I think this coin is weighted for heads”

H0: This coin is not weighted for heads.H1: This coin is weighted for heads.

Number of Heads

The formal steps of hypothesis testing:

Step 1: State your hypotheses.

Step 2: Find the critical value.

Step 3: Calculate the obtained statistic.

Step 4: Make a decision.

The Critical Value of an Inferential Statistic

Critical Value of the statistic is the value that demarcates our decision about whether something is normal vs. abnormal. (also, our decision to either support or provide support against a hypothesis).

H1: This coin is weighted for tails.

?

??

H0: This coin is not weighted for tails.

How can we determine whether or not our hypothesis is correct?

We need a rule to decide whether or not the probability of obtaining the outcome we obtained is likely to be due to chance. We need a decision rule.

An Experiment

Alpha level (): a probability level set by the investigator to delineate which outcomes will lead to supporting the alternative hypothesis. Common conventions: .05 or .01

Only 15+ tails yields a probability of .05 (without going over).So the critical value is 15.

.0370

.0148

.0046.0011

.0002.0000

.0000

H1: This coin is weighted for tails.H0: This coin is not weighted for tails.

Finding the Critical Value

= .05

Only 5 and less tails yields a probability of .05 (without going over).So the critical value is 5.

.0370

.0148

.0046.0011

.0002.0000

.0000

H1: This coin is weighted for heads.H0: This coin is not weighted for heads.

Finding the Critical Value

= .05

H1: This coin is weighted.H0: This coin is not weighted.

Finding the Critical Value

.0000.0000

.0000.0000.0002.0002

.0011.0011.0046.0046

.0148.0148

.0370.0370

5 and less or 15+ yields a probability of .05 (without going over). So the critical values are 5 and 15.

= .05

Only 16+ tails yields a probability of .01 (without going over).So the critical value is 16.

.0148

.0046.0011

.0002.0000

.0000

H1: This coin is weighted for tails.H0: This coin is not weighted for tails.

Finding the Critical Value

= .01

Only 4 and less tails yields a probability of .01 (without going over).So the critical value is 4.

.0148

.0046.0011

.0002.0000

.0000

H1: This coin is weighted for heads.H0: This coin is not weighted for heads.

Finding the Critical Value

= .01

.0046.0011

.0002.0000

.0000

H1: This coin is weighted.H0: This coin is not weighted.

Finding the Critical Value

.0000

.0000.0002

.0011.0046

3 and less or 17+ yields a probability of .01 (without going over). So the critical values are 3 and 17.

= .01

The formal steps of hypothesis testing:

Step 1: State your hypotheses.

Step 2: Find the critical value.

Step 3: Calculate the obtained statistic.

Step 4: Make a decision.

For the sign test, the obtained statistic is the result of the flip!

The question will usually give you this information.

For example: I think a coin is weighted for tails so I flip it 20 times and I get 16 tails.

Therefore, 16 tails is the obtained statistic.

The Critical Value of an Inferential Statistic

and are probabilities that correspond to different range of outcomes. They are mutually exclusive and exhaustive.

• As we increase alpha then beta must decrease and vice versa

• As we change alpha we also change the critical value & vice versa.

The formal steps of hypothesis testing:

Step 1: State your hypotheses.

Step 2: Find the critical value.

Step 3: Calculate the obtained statistic.

Step 4: Make a decision.

In Statistics, we do NOT prove ourselves right.

It is not appropriate to say “the alternative hypothesis was right.”

Instead, we use very specific terminology:

Everything is relative to the null hypothesis:

Two types of decisions or outcomes:RETAIN the null hypothesis.REJECT the null hypothesis.

The Critical Value of an Inferential Statistic

and are probabilities that correspond to different range of outcomes. They are mutually exclusive and exhaustive.

• As we increase alpha then beta must decrease and vice versa

• As we change alpha we also change the critical value & vice versa.

“rejection

region”

The Critical Value of an Inferential Statistic

and are probabilities that correspond to different range of outcomes. They are mutually exclusive and exhaustive.

• As we increase alpha then beta must decrease and vice versa

• As we change alpha we also change the critical value & vice versa.

Rejectthe Null

Retain The Null

Conclude that the coin is indeed weighted for tails.

Conclude that the coin is not weighted for tails.

A researcher is interested in determining whether or not a coin used to determine order of play during a basketball game is weighted for tails. He flips the coin 20 times. He flips the coin and gets 16 tails. Test his hypothesis. (use an alpha of .05)

Only 15+ tails will yield a p-value of .05 (without going over).

Step 1: State the null and alternative hypotheses:

Step 2: Find the critical value.

.0370

.0148

.0046.0011

.0002.0000

.0000

H0: The coin is not weighted for tails.H1: The coin is weighted for tails.

A researcher is interested in determining whether or not a coin used to determine order of play during a basketball game is weighted for tails. He flips the coin 20 times. He flips the coin and gets 16 tails. Test his hypothesis. (use an alpha of .05)

Reject the null hypothesis. The coin is weighted for tails.

Step 3: Calculate the test statistic:

Step 4: Make a decision:

16 tails

0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12

Number of Tails

Suppose it is your job to test whether a coin used at the start of a hockey game is weighted. You flip it 12 times and it comes up tails 10 times. Test the hypothesis that the coin is weighted using an alpha level of .01.

Step 1: State your null and alternative hypotheses:

Step 2: Find the critical value:

.0002.0002.0029 .0029

.0161.0161

H0: The coin is not weighted. H1: The coin is weighted.

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Number of Tails

Step 3: Calculate the obtained statistic:

Step 4: Make a decision.

.0002.0002.0029 .0029

10

We retain the null hypothesis.

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Number of Females

There are 100 students in my Statistics class: 50 females and 50 malesI have a sneaking suspicion that the brightest students are female (sorry guys!). I would like to test this hypothesis using an alpha level of .05. I make a list of the students according to their test grade and isolate the top 15 students in the class. 10 of them are females.

.0000.0005

.0032.0139

.0417

Binomial Distribution 20 events

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Number Heads

PR

obab

ility

Hypothesis testing with the Sign Test Analysis

State of RealityDecision H0 is true H0 is falseRetain H0 Correct Decision Type II error

Reject H0 Type I error Correct Decision

A decision rule based onthe probability of the outcomeor a more extreme one beingdue to chance runs the risk oftwo types of errors.